Finite Group Databasebeta

← #30670234

Subgroups — #30670234

Order 16768.

672 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862202C211yesmaximal, characteristic, complemented
2C22×(C2×D131)⋊C4C211yesmaximal, characteristic
3#22553247C212yesmaximal, complemented
4#22553241C212yesmaximal, complemented
5#22553202C212yesmaximal, complemented
6#22553201C212yesmaximal, complemented
7#22553185C211yesmaximal, characteristic, complemented
8#22553173C211yesmaximal, characteristic, complemented
9#22553167C211yesmaximal, characteristic
10#22553167C211yesmaximal, characteristic
11#385×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14#16862202—24no—
15C2×Dic131⋊C4C2211yescharacteristic, complemented
16C2×Dic131⋊C4C2211yescharacteristic, complemented
17C2×Dic131⋊C4C2212yescomplemented
18C2×C22⋊Dic131C2211yescharacteristic
19C2×C22⋊Dic131C2211yescharacteristic
20C2×(C2×D131)⋊C4—24no—
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×C4×Dic131C2211yescharacteristic, complemented
28C2×C4×Dic131C2211yescharacteristic, complemented
29C131⋊#95C2212yescomplemented
30C131⋊#95C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C23×C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2212yescomplemented
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2211yescharacteristic
36#95×C131C2212yescomplemented
37C23×D131—24no—
38C23×D131—22no—
39C23×D131—22no—
40C23×D131—22no—
41C23×D131—42no—
42C23×D131C2312yes—
43C22×Dic131—22no—
44C22×Dic131—22no—
45C22×Dic131D414yes—
46C22×Dic131C2312yes—
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2311yescharacteristic
50C22×Dic131C2311yescharacteristic
51Dic131⋊C4—22no—
52Dic131⋊C4—22no—
53Dic131⋊C4—42no—
54C22⋊Dic131—22no—
55C22⋊Dic131—22no—
56(C2×D131)⋊C4—24no—
57(C2×D131)⋊C4—24no—
58(C2×D131)⋊C4—24no—
59(C2×D131)⋊C4—24no—
60(C2×D131)⋊C4—42no—
61(C2×D131)⋊C4—42no—
62C4×Dic131—22no—
63C4×Dic131—22no—
64C24×C131C2311yescharacteristic
65C22×C4×C131—22no—
66C22×C4×C131—42no—
67C22×C4×C131C2311yescharacteristic
68C22×C4×C131C2312yes—
69C22×C4×C131C2312yes—
70C22×C4×C131C2311yescharacteristic
71C4⋊C4×C131—42no—
72C22⋊C4×C131—22no—
73C22⋊C4×C131—22no—
74C22×D131—28no—
75C22×D131—22no—
76C22×D131—22no—
77C22×D131—22no—
78C22×D131—22no—
79C22×D131—22no—
80C22×D131—22no—
81C22×D131—48no—
82C22×D131—48no—
83C22×D131—44no—
84C22×D131—44no—
85C22×D131#4114yes—
86C2×Dic131—28no—
87C2×Dic131—21no—
88C2×Dic131—21no—
89C2×Dic131—21no—
90C2×Dic131—21no—
91C2×Dic131—21no—
92C2×Dic131—21no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131—22no—
97C2×Dic131—22no—
98C2×Dic131C2×D412yes—
99C2×Dic131C2×D412yes—
100C23×C131—22no—
101C23×C131—22no—
102C23×C131C2×D412yes—
103C23×C131#4112yes—
104C23×C131#4112yes—
105C23×C131C2411yescharacteristic
106C2×C4×C131—21no—
107C2×C4×C131—21no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C2×C4×C131—44no—
117C22×C131—22no—
118C22×C131—22no—
119C22×C131—24no—
120C22×C131—24no—
121C22×C131#12314yes—
122C22×C131C22×D411yescharacteristic
123C22×C131C2×#4112yes—
124C22×C131C2×#4111yescharacteristic
125C22×C131C2×#4111yescharacteristic
126C22×C131H(5,𝔽2)12yes—
127C2×D131—24no—
128C2×D131—24no—
129C2×D131—24no—
130C2×D131—416no—
131C2×D131—42no—
132C2×D131—42no—
133C2×D131—42no—
134C2×D131—42no—
135C4×C131—41no—
136C4×C131—41no—
137C4×C131—42no—
138C4×C131—42no—
139Dic131—22no—
140Dic131—22no—
141Dic131—41no—
142Dic131—41no—
143Dic131—42no—
144C2×C131—24no—
145C2×C131C2×#12311yescharacteristic
146C2×C131#52511yescharacteristic
147C2×C131#53511yescharacteristic
148C2×C131#55112yes—
149C2×C131#55112yes—
150D131—44no—
151C131#248211yescharacteristic, complemented, Sylow
152#2482—1311nomaximal, Sylow
153C2×#117—1311no—
154C22×C22⋊C4—1311no—
155#399—1312no—
156#397—1312no—
157#388—1312no—
158#388—1312no—
159#385—1311no—
160#385D13111yescharacteristic, complemented
161#381—1311no—
162#379—1311no—
163#379—1311no—
164C25—1311no—
165C23×C4—1311no—
166C23×C4C2×D13111yescharacteristic
167#117—2624no—
168C2×C4⋊C4—1311no—
169C2×C4⋊C4—1311no—
170C2×C4⋊C4—1312no—
171C2×C4⋊C4C2×D13112yescomplemented
172C2×C22⋊C4—2624no—
173C2×C22⋊C4—1311no—
174C2×C22⋊C4—1311no—
175C2×C22⋊C4—1312no—
176C2×C22⋊C4—1312no—
177C2×C22⋊C4—1312no—
178C2×C22⋊C4—1312no—
179C2×C22⋊C4—1312no—
180C2×C22⋊C4—1312no—
181C2×C22⋊C4C2×D13111yescharacteristic
182C2×C22⋊C4C2×D13111yescharacteristic
183C2×C42—1311no—
184C2×C42—1311no—
185#95—1312no—
186#95—1312no—
187#95—1312no—
188#95C2×D13112yescomplemented
189C24—2624no—
190C24—2622no—
191C24—2622no—
192C24—2622no—
193C24—5242no—
194C24—1312no—
195C24C22×D13111yescharacteristic
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—2622no—
199C22×C4—2622no—
200C22×C4—1312no—
201C22×C4—1311no—
202C22×C4—1311no—
203C22×C4—1314no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4C22×D13112yes—
207C22×C4C22×D13112yes—
208C22×C4C22×D13111yescharacteristic
209C22×C4C22×D13111yescharacteristic
210C4⋊C4—42no—
211C4⋊C4—2622no—
212C4⋊C4—2622no—
213C4⋊C4—5242no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—2624no—
217C22⋊C4—2624no—
218C22⋊C4—2624no—
219C22⋊C4—2624no—
220C22⋊C4—2622no—
221C22⋊C4—2622no—
222C22⋊C4—5242no—
223C22⋊C4—5242no—
224C42—2622no—
225C42—2622no—
226C23—22no—
227C23—22no—
228C23—2628no—
229C23—2622no—
230C23—2622no—
231C23—2622no—
232C23—2622no—
233C23—2622no—
234C23—2622no—
235C23—5248no—
236C23—5248no—
237C23—5244no—
238C23—5244no—
239C23—1314no—
240C23D4×D13112yes—
241C23#1452351312yes—
242C23#1452351412yes—
243C23C23×D13111yescharacteristic
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—44no—
255C2×C4—2628no—
256C2×C4—2621no—
257C2×C4—2621no—
258C2×C4—2621no—
259C2×C4—2621no—
260C2×C4—2621no—
261C2×C4—2621no—
262C2×C4—2622no—
263C2×C4—2622no—
264C2×C4—2622no—
265C2×C4—2622no—
266C2×C4—2622no—
267C2×C4—1312no—
268C2×C4—1312no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—22no—
273C22—2624no—
274C22—2624no—
275C22—2624no—
276C22—52416no—
277C22—5242no—
278C22—5242no—
279C22—5242no—
280C22—5242no—
281C22#1686223114yes—
282C22C2×D4×D13111yescharacteristic
283C22#41×D13112yes—
284C22C2×#1452351311yescharacteristic
285C22C2×#1452351411yescharacteristic
286C22#1686229112yes—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—2622no—
292C4—2622no—
293C4—5241no—
294C4—5242no—
295C4—5241no—
296C2—24no—
297C2—5244no—
298C2C2×#1686223111yescharacteristic
299C2#2255382011yescharacteristic
300C2#2255387711yescharacteristic
301C2#2255398912yes—
302C2#2255399112yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#385×C131C2
prime radicalC24×D131C22
hypercenter#385D131
hypocenterC131#2482
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16862202C2semidirect
3. #22553247C2semidirect
4. #22553241C2semidirect
5. #22553202C2semidirect
6. #22553201C2semidirect
7. #22553185C2semidirect
8. #22553173C2semidirect
11. #385×C131C2semidirect
15. C2×Dic131⋊C4C22semidirect
16. C2×Dic131⋊C4C22semidirect
17. C2×Dic131⋊C4C22semidirect
27. C2×C4×Dic131C22semidirect
28. C2×C4×Dic131C22semidirect
29. C131⋊#95C22semidirect
30. C131⋊#95C22semidirect
31. C131⋊#95C22semidirect
33. C2×C4⋊C4×C131C22semidirect
36. #95×C131C22semidirect
151. C131#2482semidirect
160. #385D131semidirect
171. C2×C4⋊C4C2×D131semidirect
188. #95C2×D131semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168622022
C22×(C2×D131)⋊C42
#225532472
#225532412
#225532022
#225532012
#225531852
#225531732
#225531672
#385×C1312
#2482131